Make Airy functions symbolic

[sagetrac-olazo]

As discussed in sage-support.

Currently sage can evaluate airy functions numerically:

sage: airy_ai(1.4)
0.0820380498076

but it doesn't work symbolically

sage: airy_ai(x)
...
TypeError: Cannot evaluate symbolic expression to a numeric value.

We should make it symbolical for both airy_ai and airy_bi, as well as their derivatives.

Depends on #12289
Depends on #17130

CC: @kcrisman @burcin @benjaminfjones @fredrik-johansson @eviatarbach

Component: symbolics

Keywords: Airy functions sd40.5 sd48

Author: Oscar Gerardo Lazo Arjona, Benjamin Jones, Douglas McNeil, Eviatar Bach, Ralf Stephan

Branch: 2f6945a

Reviewer: Eviatar Bach, Karl-Dieter Crisman, Burcin Erocal, Ralf Stephan, Jeroen Demeyer, Marc Mezzarobba

Issue created by migration from https://trac.sagemath.org/ticket/12455

[sagetrac-olazo]

comment:4

Attachment: trac_12455-symbolic_airy.patch.gz

I've added a patch, which should do the job, but it has a few shortcomings:

1.-The resulting symbolic functions seem to remain on hold:

sage: airy_ai(1.0)
airy_ai(1.00000000000000)

You need to force it to evaluate:

sage: airy_ai(1.0).n()
0.135292416313

2.- This doesn't work:

sage: airy_ai(2.0).n(digits=100)
0.0349241304233

3.- There is no evaluation for airy_ai_prime or airy_bi_prime

4.- I'm not sure about how should the functions be called, some possible schemes are

{ai,bi,aip,bip}
{ai,bai,aip,baip}
{airy_ai,airy_bi,airy_ai_prime,airy_bi_prime}

And also whether the latex representation should be capitalized or not. I chose the third scheme, and capitalized typesetting.

[kcrisman]

comment:6

Good start. A few responses.

• You'll need to have more methods. We should probably use mpmath to evaluate these functions in general, though using the SciPy version could be useful over RDF. See Sage is missing the lambert_w function #11888 for a very recent example of this kind of thing and how to implement it.
• That should also fix precision issues, if properly done, I think.
• SciPy has evaluation for the airy primes. I think that Maxima has these too. (Cc:ing FJ in case he has plans to add these to mpmath.)
• We should have translations of these to Maxima and friends. That will probably help us name them. What do Mathematica and Maple call them?
• All methods, including initialization methods, need doctests. Many of the new symbolic functions added at symbolics/functions will have nice models to follow.

Thanks for getting us a start on this! That's great.

[fredrik-johansson]

comment:7

mpmath has the derivatives (as well as integrals) -- just use the optional 'derivative' parameter.

[kcrisman]

comment:8

mpmath has the derivatives (as well as integrals) -- just use the optional 'derivative' parameter.

Ok, I didn't realize that was what that parameter was about. Though in retrospect it seems obvious!

[sagetrac-olazo]

comment:9

On second thought, I think it would be better to use the airy equation to calculate derivatives or order higher than 1. Like

sage: airy_ai(2,x)
x*airy_ai(x)
sage: airy_ai(3,x)
airy_ai(x)+x*airy_ai_prime(x)
sage: diff(airy_ai(x),x,2)
x*airy_ai(x)
sage: diff(airy_ai(x),x,3)
airy_ai(x)+x*airy_ai_prime(x)

which is very likey to be the way mpmath calculates higher order derivatives. Integrals however, would be returned as:

sage: airy_ai(-1,x)
airy_ai(-1,x)
sage: integral(airy_ai(x),x)
airy_ai(-1,x)

what do you think?

[sagetrac-olazo]

comment:10

Attachment: trac_12455-symbolic_airy2.patch.gz

I just added a new patch. The new version includes generalized derivatives, evaluation with mpmath, and special values of the functions and their derivatives. Just a one doubt, the coverage is:

oscar@oscar-netbook:~$ sage -coverage airy.py
SCORE /home/oscar/sage/my_patches/airy.py: 76% (20 of 26)

Missing documentation:
	 * __init__(self):
	 * __init__(self):
	 * __init__(self):
	 * __init__(self):
	 * __init__(self):
	 * __init__(self):


Possibly wrong (function name doesn't occur in doctests):
	 * _derivative_(self, x, diff_param=None):
	 * _eval_(self, x):
	 * _evalf_(self, x, parent=None):
	 * _derivative_(self, x, diff_param=None):
	 * _eval_(self, x):
	 * _evalf_(self, x, parent=None):
	 * _derivative_(self, alpha, *args, **kwds):
	 * _eval_(self, alpha, *args):
	 * _evalf_(self, alpha, x, parent=None):
	 * _derivative_(self, x, diff_param=None):
	 * _eval_(self, x):
	 * _evalf_(self, x, parent=None):
	 * _derivative_(self, x, diff_param=None):
	 * _eval_(self, x):
	 * _evalf_(self, x, parent=None):
	 * _derivative_(self, alpha, *args, **kwds):
	 * _eval_(self, alpha, *args):
	 * _evalf_(self, alpha, x, parent=None):

Is that important?

[kcrisman]

comment:11

Is that important?

The second part isn't, as you are implicitly testing these 'hidden' functions. By the way,

Examples::

should be

EXAMPLES::

However, the first part (the initialization methods) is important for our coverage. You can just do a couple from the big examples that you have.

Do these pass doctests? I'm surprised that

sage: airy_ai_simple= FunctionAiryAiSimple()

would work when you run tests, assuming you didn't import FunctionAiryAiSimple into the global namespace.

Anyway, overall on a quick glance this looks great; unfortunately, I won't have time to review it properly soon - hopefully someone else will, because you put a lot of great work into it and we should totally have these symbolic. Thanks!

[sagetrac-olazo]

comment:12

Do these pass doctests? I'm surprised that

sage: airy_ai_simple= FunctionAiryAiSimple()

would work when you run tests, assuming you didn't import FunctionAiryAiSimple into the global namespace.

It looks like they do:

oscar@oscar-netbook:~$ sage -coverage airy.py
sage -t  "/home/oscar/sage/my_patches/airy.py"              
	 [51.5 s]
 
----------------------------------------------------------------------
All tests passed!
Total time for all tests: 51.7 seconds

Why shouldn't they? I'll take care of the __init__ functions.

[sagetrac-olazo]

comment:13

I'm sorry, that should be:

oscar@oscar-netbook:~$ sage -t airy.py
sage -t  "/home/oscar/sage/my_patches/airy.py"              
	 [51.5 s]
 
----------------------------------------------------------------------
All tests passed!
Total time for all tests: 51.7 seconds

(the command was sage -t)

[kcrisman]

comment:14

sage: airy_ai_simple= FunctionAiryAiSimple()

would work when you run tests, assuming you didn't import FunctionAiryAiSimple into the global namespace.

I'm just surprised this doesn't cause trouble.

On another note, this apparently does something weird when applied to 5.0.beta4. Namely, Sage won't start:

---> 65 from sage.functions.all import sin, cos

ImportError: cannot import name sin
Error importing ipy_profile_sage - perhaps you should run %upgrade?
WARNING: Loading of ipy_profile_sage failed.

I'll spare you the rest of the traceback. I think there is some kind of circular import thing going on here, but I don't understand imports that well. Perhaps moving the import of airy to further down (after trig functions) would help - that's a totally uninformed guess, of course.

[sagetrac-olazo]

comment:15

Attachment: trac_12455-symbolic_airy3.patch.gz

I just added a new patch, now with 100 % test coverage, but I still get this message:

oscar@oscar-netbook:~$ sage -coverage /home/oscar/sage/my_patches/airy.py ----------------------------------------------------------------------
/home/oscar/sage/my_patches/airy.py
ERROR: Please add a `TestSuite(s).run()` doctest.
SCORE /home/oscar/sage/my_patches/airy.py: 100% (26 of 26)
----------------------------------------------------------------------

What is this TestSuite thing? I also changed the order in which functions are initialized, so that airy_ai_general is initialized first. Doing it last made some symbolic operations, such as simplify not work (It said something about airy_ai requiring 2 arguments).

[sagetrac-olazo]

Author: Oscar Gerardo Lazo Arjona

[sagetrac-git]

Changed commit from aac86a2 to 070e728

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

3cac775	Merge branch 'develop' into t/12455/public/ticket/12455
070e728	12455: doc typos

[mezzarobba]

comment:78

x should be z in the differential equation

[mezzarobba]

comment:79

What exactly still needs review here?

The code looks pretty good to me overall, but there are lots of things I don't understand in detail. (For a simple example, what is the motivation for splitting the implementation in so many unrelated classes?) And yet I'm tempted to give this ticket positive review (leaving possible remaining issues for later), since many other people have looked at it and it is clearly an improvement over what sage currently has.

[mezzarobba]

comment:80

More nitpicking, not necessarily for this ticket:

• Raising a ValueError in FunctionAiryAiGeneral._derivative_ when diff_param == 0 may not be the most appropriate, since (if I'm not mistaken) the partial derivative would make sense mathematically.
• Is it necessary to special-case alpha == 0 etc. in FunctionAiryAiGeneral._evalf_?

[mezzarobba]

comment:81

Replying to @mezzarobba:

• Is it necessary to special-case alpha == 0 etc. in FunctionAiryAiGeneral._evalf_?

Scratch that, I misread the code.

[sagetrac-git]

Changed commit from 070e728 to 91530d0

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

91530d0

Ai/Bi: doc fixes and clarifications

[mezzarobba]

comment:83

One last thing: IMO hold_derivative should be True by default in airy_ai and airy_bi, except perhaps when the differentiation order is one. But I don't know if that's consistent with the way things are done in other parts of sage symbolics. Thoughts?

[rwst]

comment:84

Replying to @mezzarobba:

x should be z in the differential equation

Strange typo.

• Raising a ValueError in FunctionAiryAiGeneral._derivative_ when diff_param == 0 may not be the most appropriate, since (if I'm not mistaken) the partial derivative would make sense mathematically.

A quick look at old and new code shows 1. ValueError, 2. NotImplementedError, 3. assert. So, this needs to be unified to NotImplementedError.

One last thing: IMO hold_derivative should be True by default in airy_ai and airy_bi, except perhaps when the differentiation order is one. But I don't know if that's consistent with the way things are done in other parts of sage symbolics. Thoughts?

I believe the code of this ticket is the only one that has such a parameter, so please feel free to improve it.

I have looked at your patch and it's fine, so if you think the rest is OK please set positive.

[sagetrac-git]

Changed commit from 91530d0 to 2f6945a

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

347cd08	#12455 Airy functions: doc fixes and clarifications
6aaf2da	#12455 Airy funs: raise NotImplementedError on diff(airy_?i(α,x),α)
c80be63	#12455 Airy funs: hold_derivative by default
2f6945a	#12455 Airy functions: avoid calling deprecated function

[mezzarobba]

comment:86

Replying to @rwst:

• Raising a ValueError in FunctionAiryAiGeneral._derivative_ when diff_param == 0 may not be the most appropriate, since (if I'm not mistaken) the partial derivative would make sense mathematically.

A quick look at old and new code shows 1. ValueError, 2. NotImplementedError, 3. assert. So, this needs to be unified to NotImplementedError.

One last thing: IMO hold_derivative should be True by default in airy_ai and airy_bi, except perhaps when the differentiation order is one. But I don't know if that's consistent with the way things are done in other parts of sage symbolics. Thoughts?

I believe the code of this ticket is the only one that has such a parameter, so please feel free to improve it.

Both done (now rather than later to avoid breaking the interface).

I have looked at your patch and it's fine, so if you think the rest is OK please set positive.

I'm okay with everything prior to my commits. Could you (or someone else) have a quick look at my last changes and set the ticket to positive review? Thanks!

[mezzarobba]

Changed reviewer from Eviatar Bach, Karl-Dieter Crisman, Burcin Erocal, Ralf Stephan to Eviatar Bach, Karl-Dieter Crisman, Burcin Erocal, Ralf Stephan, Jeroen Demeyer, Marc Mezzarobba

[rwst]

comment:87

Is fine and passes tests in functions/. Thank you too.

[vbraun]

Changed branch from public/ticket/12455 to 2f6945a

[kcrisman]

Changed author from Oscar Gerardo Lazo Arjona, Benjamin Jones, D. S. McNeil, Eviatar Bach, Ralf Stephan to Oscar Gerardo Lazo Arjona, Benjamin Jones, Douglas McNeil, Eviatar Bach, Ralf Stephan

[kcrisman]

Changed commit from 2f6945a to none